Equidistribution theorems for holomorphic Siegel cusp forms of general degree: the level aspect

Henry H. Kim, Satoshi Wakatsuki, Takuya Yamauchi

Published 2021-06-15Version 1

We prove equidistribution theorems for a family of holomorphic Siegel cusp forms of general degree in the level aspect. Our main contribution is to estimate unipotent contributions for general degree in the geometric side of Arthur's invariant trace formula in terms of Shintani zeta functions. Several applications including the vertical Sato-Tate theorem and low-lying zeros for standard $L$-functions of holomorphic Siegel cusp forms are discussed. As a byproduct, we also give a higher dimensional generalization of a theorem of Brumer. We also show that the ``non-genuine forms" which come from non-trivial endoscopic contributions by Langlands functoriality classified by Arthur are negligible in the level aspect.